Question: Simplify the following expression: $r = \dfrac{-49a^2 + 70a}{56a^2 + 7a}$ You can assume $a \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-49a^2 + 70a = - (7\cdot7 \cdot a \cdot a) + (2\cdot5\cdot7 \cdot a)$ The denominator can be factored: $56a^2 + 7a = (2\cdot2\cdot2\cdot7 \cdot a \cdot a) + (7 \cdot a)$ The greatest common factor of all the terms is $7a$ Factoring out $7a$ gives us: $r = \dfrac{(7a)(-7a + 10)}{(7a)(8a + 1)}$ Dividing both the numerator and denominator by $7a$ gives: $r = \dfrac{-7a + 10}{8a + 1}$